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Groups satisfying Gaschütz's complement theorem
Abstract
A classical theorem of Gaschütz states that an abelian normal subgroup N of a finite group G has a complement in G whenever it has a complement in some subgroup H with N ≤ H ≤ G and gcd(|N|, [G:H]) = 1. We characterize the finite groups N for which this theorem remains valid without the abelian hypothesis: this is the case if and only if Z(N) ∩ N' = 1 and N has a complement in a quotient of the holomorph Hol(N) by a diagonal copy of N'. This completes a previous paper by the second author.
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BibTeX
@misc{aivazidis2026groups,
title = {{Groups satisfying Gaschütz's complement theorem}},
author = {Stefanos Aivazidis and Benjamin Sambale},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27579},
eprint = {2608.27579},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27579},
}
RIS
TY - UNPB
TI - Groups satisfying Gaschütz's complement theorem
AU - Stefanos Aivazidis
AU - Benjamin Sambale
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27579
UR - https://arxiv.org/abs/2608.27579
AB - A classical theorem of Gaschütz states that an abelian normal subgroup N of a finite group G has a complement in G whenever it has a complement in some subgroup H with N ≤ H ≤ G and gcd(|N|, [G:H]) = 1. We characterize the finite groups N for which this theorem remains valid without the abelian hypothesis: this is the case if and only if Z(N) ∩ N' = 1 and N has a complement in a quotient of the holomorph Hol(N) by a diagonal copy of N'. This completes a previous paper by the second author.
ER -